Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence
A Constructive Argumentation Framework
Souhila Kaci
Yakoub Salhi
LIRMM - UMR 5506, University of Montpellier 2
161 rue ADA F34392 Montpellier Cedex 5, France
kaci@lirmm.fr
CRIL - UMR 8188, University of Artois
F-62307 Lens Cedex, France
salhi@cril.fr
Abstract
present paper follows the second trend. In particular we focus on the case where the arguments are built from a set
of classical propositional logic formulas. Therefore an argument is seen as a reason for or against the truth of a particular statement. In particular, an argument is a pair of (1)
a set of classical propositional formulas obeying some conditions, called the support of the argument, and (2) a logical
formula inferred from the set, called the conclusion of the argument. Given a set of arguments (constructed from a set of
classical propositional formulas), the classical treatment of
argumentation framework consists in determining conflicts
over the arguments and determining acceptable arguments.
Conclusions supported by acceptable arguments are consistent and considered reliable. Although works around argumentation frameworks have shown great promises for reasoning with inconsistent knowledge, the argumentation approach also has important shortcomings in this setting. More
precisely, in some applications what one is interested in are
not so much only the conclusions supported by the acceptable arguments but also the explications of such conclusions.
Unfortunately, in some situations the internal structure of the
acceptable arguments is not sufficient to provide such information. Let us consider the following example to illustrate
the problem:
• If John has taken his medication (M ), then his situation is
stable (S).
• If John has not taken his medication, then a doctor administers the medication to him (D).
• If a doctor administers the medication to John, then the
situation of John is stable.
From the previous statements, we can deduce that the situation of John is stable (S) using classical reasoning. Indeed,
this can be easily obtained using the law of excluded middle
(M ∨ ¬M ): on one hand, if M is true then S is true. On
the other hand, if ¬M is true then D is true which allows
us to conclude that S is true too. It goes without saying that
this situation is extremely confusing as, although we know
that the situation of John is stable, we are not able to provide the precise reason for such a conclusion if we don’t
know whether John has taken his medication or not! For instance, if John lost consciousness, we, including the doctor,
cannot determine which of M or ¬M is true in M ∨ ¬M .
Clearly we need an argumentation framework over a logic
Dung’s argumentation framework is an abstract framework based on a set of arguments and a binary attack
relation defined over the set. One instantiation, among
many others, of Dung’s framework consists in constructing the arguments from a set of propositional logic
formulas. Thus an argument is seen as a reason for or
against the truth of a particular statement. Despite its
advantages, the argumentation approach for inconsistency handling also has important shortcomings. More
precisely, in some applications what one is interested in
are not so much only the conclusions supported by the
arguments but also the precise explications of such conclusions. We show that argumentation framework applied to classical logic formulas is not suitable to deal
with this problem. On the other hand, intuitionistic logic
appears to be a natural alternative candidate logic (instead of classical logic) to instantiate Dung’s framework. We develop constructive argumentation framework. We show that intuitionistic logic offers nice and
desirable properties of the arguments. We also provide
a characterization of the arguments in this setting in
terms of minimal inconsistent subsets when intuitionistic logic is embedded in the modal logic S4.
Introduction
Argumentation theory is a reasoning process based on constructing arguments, determining conflicts between arguments and determining acceptable arguments. Dung’s argumentation framework is an abstract framework based on a
set of arguments and a binary attack relation defined over the
set (Dung 1995). In this framework, an argument is an abstract entity whose origin and structure are not known. The
role of an argument is only described by its conflicts with
other arguments. The abstract nature of Dung’s framework
accounts for the broad range of its applications.
We distinguish between two main trends for extending/instantiating Dung’s framework: those argumentation
frameworks which use (as in Dung’s framework) abstract
arguments, e.g. (Bench-Capon 2003), and those which take
into account the internal structure of the arguments, e.g.,
(Simari and Loui 1992; Besnard and Hunter 2008). The
c 2014, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
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w M p iff p ∈ V (w);
which, contrary to classical logic, goes beyond the classical
deduction of the conclusion of an argument from the support
of the argument. This consideration calls for more sophisticated logics. In other words, what we need is not only the
information from which a certain conclusion is derived but
also that information which permit to construct the conclusion at hand. This consideration is formally known in the
literature as “constructivism”.
In mathematics, constructivism is a philosophical approach which consists in requiring that in order to prove
that a mathematical object exists, it is necessary to be
able to construct it. In this context, intuitionistic propositional logic can be seen as a formalization of constructive mathematics of Brouwer and Heyting (Beeson 1984;
Troelstra and van Dalen 1988). It is defined starting from
classical logic by excluding certain standard forms of reasoning, in particular, the law of excluded middle and doublenegation elimination. A statement in intuitionistic logic is
considered as true if we are able to provide a constructive
proof, and as false if we are able to prove that it does not
have a constructive proof. In a sense, the truth value of a
statement is more related to our knowledge than in classical
logic. For instance, the formula M ∨ ¬M is true in intuitionistic logic only if we provide a constructive proof of M or
a constructive proof of ¬M , whereas this formula is true in
classical logic without such proofs. Thus, if we consider the
previous example, it is not possible to derive S in intuitionistic logic from the given statements.
In the setting of argumentation framework for handling
inconsistent logical formulas, intuitionistic logic appears to
be a suitable tool to address the above considerations. In this
paper, we develop a constructive argumentation framework
which builds on Dung’s framework and intuitionistic logic.
w M ⊥ never holds;
w M A ∧ B iff w M A and w M B;
w M A ∨ B iff w M A or w M B;
w M A → B iff, for all w0 ∈ W with w w0 , if
w0 M A then w0 M B.
Note that M satisfies Kripke monotonicity property:
Proposition 1. Let A be a formula, M = (W, , V ) and
w, w0 ∈ W . If w M A and w w0 , then w0 A.
A formula A is satisfiable in IPL if there exists a Kripke
model M = (W, , V ) and a world w in W such that w M
A. A is valid in IPL if, for all Kripke model M = (W, , V )
and for all w in W , w M A. Satisfiability and validity in
IPL are Polynomial-Space Complete (Statman 1979).
For logical consequence concept, we use the turnstile
symbol ”`i ” (i for intuitionistic propositional logic), i.e.,
{A1 , . . . , An } `i B and (A1 ∧ · · · ∧ An ) → B are equivalent. Similarly to classical logic, we have the deduction
property in intuitionistic logic:
Proposition 2. Let Γ be a set of formulæ, and A and B two
formulæ. Then, Γ `i A → B iff Γ ∪ {A} `i B.
Classical logic is stronger than intuitionistic logic. Therefore a valid formula in the latter is also valid in the former.
Classical vs Constructive Arguments
In this section, we first present the usage of Dung’s framework for handling inconsistency in classical propositional
logic knowledge bases (Besnard and Hunter 2008). We then
discuss how this instantiation can or can not be directly applicable with intuitionistic propositional logic. Essentially,
a logic-based argumentation framework operates in the following steps:
Intuitionistic Propositional Logic
We first define the syntax of intuitionistic propositional logic
(IPL for short). Let Prop be a denumerable set of propositional variables whose elements are denoted by p, q, r, etc.
The symbols ∨, ∧, → denote disjunction, conjunction and
implication, respectively. The set of formulæ of IPL, denoted
Form, is defined from Prop and the constant ⊥, denoting absurdity, by using the following grammar:
A ::= p | ⊥ | A ∧ A | A ∨ A | A → A.
The logical connective of negation ¬ is defined by using ⊥
and the connective →: ¬A ≡ A → ⊥. The logical connective of equivalence, denoted ↔, is defined as usual.
We provide here a possible world semantics, called
Kripke semantics, of intuitionistic logic. In this semantics,
we use a universe of worlds where each propositional variable has a truth value in each world.
Definition 1 (Kripke Model). Let W be the universe of
worlds. A Kripke model is defined as a triple (W, , V ),
where is a preorder over W and V : W → 2Prop is an interpretation such that, for all w and w0 in W with w w0 ,
V (w) ⊆ V (w0 ).
We associate to each Kripke model M = (W, , V ) a
forcing relation, denoted M , between W and Form. It is
defined by induction on formula structure as follows:
1. constructing arguments (in favor of/against a conclusion)
from knowledge bases,
2. determining the conflicts, called an attack relation, between the arguments,
3. and determining the acceptable arguments from which
justified conclusions are concluded.
Dung’s argumentation framework is a pair hA, Atti, where
A is the set of arguments and Att is the attack relation over
A×A. Acceptability semantics define sets of arguments that
should satisfy some conditions in order to represent a justifiable point of view on the acceptance of the arguments. Due
to the lack of space we do not recall these semantics and
refer the reader to (Dung 1995). The notion of argument is
defined on the basis of the underlying logical language and
its associated logical consequence.
Classical Argumentation Framework
When Dung’s argumentation framework is used to deal with
inconsistent knowledge encoded in classical propositional
logic, an argument is defined in the following way:
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Definition 2 (Argument). Let Γ be a set of classical propositional formulas. An argument over Γ is a pair A = h∆, Ci
such that ∆ ⊆ Γ, ∆ 0c ⊥, ∆ `c C (c for classical propositional logic) and, for all ∆0 ⊂ ∆, ∆0 0c C.
have a strong background on IPL. What they need to know
in this section are the following properties2 : (i) {¬¬A} `i A
does not hold, (ii)A → ¬¬A is a theorem of IPL and
(iii) A ∨ ¬A is not a theorem of IPL.
With this in mind, we can first state that:
a classical argument is not necessarily a constructive argument.
For example h{A → C, ¬A → B, B → C}, Ci is a classical argument but not a constructive one.
Moreover the characteristic property of a classical argument stating that h∆, Ci is a classical argument if and
only if ∆ ∪ {¬C} is a minimal inconsistent subset does
not hold with constructive arguments. For instance, the set
{¬¬p, ¬p} is a minimal inconsistent subset, but h{¬¬p}, pi
is not a constructive argument. This is because of {¬¬p} 0i
p. Indeed, the formula ¬¬p → p has as a counter-model
M = ({w, w0 }, , V ) with w w0 , V (w) = ∅ and
V (w0 ) = {p}. In this Kripke model, we have w M ¬¬p,
since w0 M p, and w 2M p.
However, we have the following property:
The set ∆ is called the support of the argument and
C its conclusion. We say that A is an argument for C.
In this setting, h∆, Ci is called a classical argument. For
instance, consider Γ = {A → C, ¬A → B, B → C}. The
pair h∆, Ci, with ∆ = Γ, is a classical argument. Given
two arguments h∆, Ci and h∆0 , C 0 i, we say that h∆, Ci
undercuts h∆0 , C 0 i iff for some φ ∈ ∆0 , C ≡ ¬φ. h∆, Ci
rebuts h∆0 , C 0 i iff C ≡ ¬C 0 . Then, h∆, Ci attacks h∆0 , C 0 i
iff h∆, Ci rebuts/undercuts h∆0 , C 0 i1 .
The practical computation of the arguments is far from
being a straightforward problem. Given the setting of classical propositional logic and the properties of an argument, the
authors of (Besnard et al. 2010) have proposed an approach
based on insights from SAT problem (Grégoire, Mazure, and
Piette 2009). More precisely, let h∆, Ci be a classical argument. From Definition 2 we have that ∆ is minimal. Moreover we have ∆ `c C which is equivalent to ∆ ∪ {¬C} being inconsistent, i.e., ∆ ∪ {¬C} `c ⊥. Therefore ∆ ∪ {¬C}
is minimally inconsistent. The subset is inconsistent but all
its proper subsets are consistent. From the previous equivalence, a classical argument is characterized in the following
way (Besnard and Hunter 2008):
h∆, Ci is a classical argument if and only if ∆ ∪ {¬C} is
a minimal inconsistent subset.
In SAT terminology, a minimal inconsistent subset is
called MUS (for Minimally Unsatisfiable Subset). An algorithm is provided in (Besnard et al. 2010) to generate the set
of arguments in an efficient way.
Proposition 3. If h∆, Ci is a constructive argument, then
∆ ∪ {¬C} is an inconsistent subset.
Proof. We have ∆ `i C, since the pair h∆, Ci is a constructive argument. Moreover, we have C `i ¬¬C (C → ¬¬C is
a valid formula in IPL). Thus, using ∆ `i C and C `i ¬¬C,
we obtain ∆ `i ¬¬C. From Prop. 2, we deduce that ∆ ∪
{¬C} is an inconsistent set, since ¬¬C ≡ ¬C → ⊥.
The inconsistent subset in the previous proposition is not necessarily minimal. Indeed, the pair
h{¬¬p, ¬¬p → p}, pi is a constructive argument, but
{¬¬p, ¬¬p → p, ¬p} is not a minimal inconsistent set
because {¬¬p, ¬p} is an inconsistent set.
Nonetheless, constructive arguments can be characterized
by means of minimal inconsistent subsets when the conclusion of the argument has the form ¬C. Formally, we have:
Constructive Argumentation Framework
Let us now focus our attention on our main problem, namely
argumentation for intuitionistic propositional logic. We define a constructive argumentation framework as an instantiation of Dung’s framework over intuitionistic propositional
logic. It also operates in the three steps previously described.
The main difference however consists in the logical consequence. More precisely due to the use of intuitionistic
(instead of classical) propositional logic, the logical consequence `c in Definition 2 is replaced with `i . The argument
is then called constructive. The definitions of the attack relation and acceptability semantics remain unchanged.
Having defined what constructive argumentation framework is, we naturally come to the question “How do classical and constructive argumentation frameworks relate to
each other?” While the definition of the attack relation and
acceptability semantics are identical, we will show that the
notion of argument and its characterization with a minimally
inconsistent subset fall down in the setting of intuitionistic propositional logic. For this purpose, readers need not to
Proposition 4. The pair h∆, ¬Ci is a constructive argument
iff ∆ ∪ {C} is a minimal inconsistent subset.
Proof.
Part ⇒. Using Prop. 2, we have if ∆ `i ¬C then ∆∪{C} `i
⊥. Hence, ∆ ∪ {C} is an inconsistent subset. If ∆ ∪ {C}
is not a minimal inconsistent subset then there exists ∆0 ⊂
∆ such that ∆0 `i ¬C. We get a contradiction since ∆ is
minimal from Def. 2, i.e., no proper subset of ∆ deduces
¬C. Therefore, ∆ ∪ {C} is a minimal inconsistent subset.
Part ⇐. If ∆∪{C} is a minimal inconsistent subset, then we
have ∆ 0i ⊥, ∆ `i ¬C (Prop. 2) and, for all ∆0 ⊂ ∆, ∆0 0i
¬C. Thus, the pair h∆, ¬Ci is a constructive argument.
Proposition 4 comes from the fact that constructing the
negation of a formula corresponds to having a contradiction
(⊥) from this formula considered as an hypothesis, since ¬C
is seen as the formula C → ⊥ in IPL.
Lastly, let us emphasize that a constructive argument can
be seen as a classical argument augmented with additional
1
In some argumentation systems, called preference-based argumentation frameworks, the attack relation is derived from a conflict
relation and a preference relation over the arguments. However this
is not the main focus in this paper.
2
A formal exposition of IPL and its consequence in argumentative reasoning will be given in the next section.
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∧, ∨ and →. The most of de Morgan laws are not valid in
this logic. For instance, the formula A → B is not equivalent to ¬A ∨ B. Moreover, double-negation elimination is
excluded from IPL, i.e., A is not equivalent to ¬¬A.
Intuitively reasoning in intuitionistic logic is not carried
out in terms of ”true” and ”false”, but in terms of ”proof” and
”contradiction”. For instance, the formula ¬(A ∧ B) means
that we can prove from both A and B that we get a contradiction. However it does not mean that we can prove that
(i) we get a contradiction from A or (ii) we get a contradiction from B as it may be suggested by (¬A ∨ ¬B).
For instance, consider the formula ¬(p ∧ q) → (¬p ∨ ¬q)
which is valid in classical logic. This formula is not
valid in intuitionistic logic because it admits the countermodel M = ({w, w0 , w00 }, , V ) where is defined by
w w0 and w w00 , and V (w) = ∅, V (w0 ) = {p} and
V (w0 ) = {q}. Indeed, we have w M ¬(p ∧ q), because of
w0 2M p ∧ q and w00 2M p ∧ q; and we have w 2M ¬p
because of w0 M p, and w 2M ¬q because of w00 M q.
information that allow us to comply with the principles of
constructivism.
Proposition 5. If h∆, Ci is a constructive argument, then
there exists ∆0 ⊆ ∆ s.t. h∆0 , Ci is a classical argument.
The proof of the previous proposition relies on the fact
that each valid formula in IPL is also valid in classical logic.
Besides the fact that constructive arguments provide a
way to construct a given conclusion, they also prevent
some undesirable justifications when classical arguments are
dealt with. By undesirability we mean here that the justification is misleading. In order to illustrate this point, we
consider a simple medical diagnostic framework. We use
rules of the form S1 ∧ · · · ∧ Sn → D, where S1 , . . . , Sn
denote symptoms and D a disease. A diagnosis consists
in searching for a rule for which the left part matches
symptoms of a patient. Consider the following rules and
symptoms of a patient: S1 → D, (¬S1 ∧ S2 ) → D,
S1 , S2 . We fix Γ as the set of the previous formulæ. The
pair A = h{S1 → D, (¬S1 ∧ S2 ) → D, S2 }, Di is a classical argument over Γ. Clearly this argument cannot be considered as such. This is because it is based on the law of
excluded middle. Indeed, by using the validity of S1 ∨ ¬S1 ,
if S1 then D is true because of the rule S1 → D; otherwise, ¬S1 is true and, by using the rule (¬S1 ∧ S2 ) → D,
D is also true, since we have S2 in the support. Therefore
this argument suggests that D is true because S1 is true or
¬S1 ∧ S2 is true. However only S1 is true and ¬S1 ∧ S2 cannot by no mean considered as a possible justification of D.
The argument A is not constructive because it is based on
the law of excluded middle. There exists a single constructive argument over Γ having D as conclusion which is the
pair h{S1 , S1 → D}, Di. This argument provides the justification explaining why the patient has the disease D, i.e.,
the patient has the symptom S1 . Indeed we can say that constructive arguments get rid of imprecision (S1 or ¬S1 ∧ S2 )
and provide arguments with a precise justification, namely
S1 in the previous example.
Let us consider the following statements:
1. Peter cannot be an owner and a tenant of an apartment:
¬(O ∧ T ).
2. Peter is an owner of an apartment: O.
We put Γ = {¬(O ∧ T ), O}. The pair A = h{¬(O ∧
T )}, ¬O ∨ ¬T i is a classical argument over Γ. This means
that from ¬(O ∧ T ) we obtain that Peter is not an owner
of an apartment or he is not a tenant of an apartment. Note
that the conclusion of A is obtained by using one of the following instances of the law of excluded middle: (O ∨ ¬O)
and (T ∨ ¬T ). Indeed, if we have O (resp. T ) then, by using ¬(O ∧ T ), we have ¬T (resp. ¬O). Otherwise, we have
¬O (resp. ¬T ). Thus, the argument A allows us to know
¬O ∨ ¬T without precisely knowing whether Peter is not an
owner of an apartment (¬O) or he is not a tenant of an apartment (¬T ). The unique constructive argument over Γ having
¬O ∨ ¬T as conclusion is h{O, ¬(O ∧ T )}, ¬O ∨ ¬T i. This
argument is constructive because we know that Peter is not
a tenant (¬T ) and, a fortiori, we have ¬O ∨ ¬T .
As a second example, consider the formula (p → q) →
(¬p ∨ q) which is not valid in intuitionistic logic. A countermodel, among others, of this formula is M = ({w, w0 }, , V ) where V (w) = ∅, V (w0 ) = {p, q} and is defined by
w w0 . This is obtained from w M p → q, w 2M ¬p
because of w0 M p, and w 2M p.
In order to illustrate the fact that the formula (p → q) →
(¬p ∨ q) is not valid in intuitionistic logic, consider the following statements:
Properties of Intuitionistic Logic: Application
to Constructive Arguments
We have shown in the previous section that instantiating
Dung’s framework with intuitionistic logic defines a new
argumentation framework in which the notion of argument
and its characterization with a minimal inconsistent set completely differs from classical argumentation framework. Not
only does the new framework compute an argument in favor
of a statement but it also builds the precise justification for
that statement. Given the virtues of IPL, in this section we
go into a more detailed exposition of this logic. We illustrate
its applicability on argumentation reasoning.
• If Peter is a tenant, then he will buy an apartment: T → B.
• Peter is a tenant: T .
We put Γ = {T → B, T }. In the classical argument h{T →
B}, ¬T ∨ Bi we know that Peter is not a tenant or he will
buy an apartment without exactly knowing which of these
statements is true. In this context, constructivism requires to
know this information in order to obtain the conclusion ¬T ∨
B. For instance, h{T, T → B}, ¬T ∨ Bi is a constructive
Non-Interdefinability of Connectives
In classical logic, it is possible to define the logical connective ∧ (resp. ∨) by using ∨ and ¬ (resp. ∧ and ¬) which
is not possible in intuitionistic logic. Indeed in intuitionistic
logic, it is not possible to reformulate the logical connectives
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Γ, A ` A
[Id]
Γ, ⊥ ` C
Γ, A, B ` C
Γ, A ∧ B ` C
Γ ` Ai
Γ`A
[∧L ]
Γ`B
Γ`A∧B
i∈{1,2}
Γ ` A1 ∨ A2
[∨R
Γ, A ` C
]
1996)). A sequent S has a proof in GIPL if it has a finite
derivation in GIPL where each leaf node is labeled with
an axiom. For instance, we provide a proof of {p → r,
q → r} ` p ∨ q → r using GIPL in Figure 2.
We now show a property satisfied by constructive arguments that comes from the disjunction property.
[⊥]
[∧R ]
Γ, B ` A
Γ, A ∨ B ` C
[∨L ]
Proposition 6. If h∆, A ∨ Bi is a constructive argument
and ∨ does not appear in ∆, then h∆, Ai or h∆, Bi is a
constructive argument.
Γ, A ` B
Γ`A→B
Γ, A → B ` A
[→R ]
∆, B ` C
Γ, ∆, A → B ` C
[→L ]
Proof. By induction on the proof of ∆`A∨B in the sequent
calculus GIPL . Note that ∆ ` A ∨ B can not be an instance of
an axiom ([⊥] and [Id]), since ∆ 0 ⊥ and A∨B is not a subformula of ∆. If the last application rule is a right rule, then
it is an instance of [∨R ]. Hence, we have a proof of ∆ ` A
or ∆ ` B in GIPL . Consequently, h∆, Ai is a constructive
argument or h∆, Bi is a constructive argument, since ∆ 0 ⊥
and ∆ is minimal. We now consider the case where the last
application rule is an instance of a left rule. In this case, the
last application rule is an instance of either [∧L ] or [→L ].
In the case of [∧L ]:
Figure 1: Sequent Calculus GIPL
argument because its support allows us to know that Peter
will buy an apartment.
As a third example, consider the following statement: it is
note true that if the suspect is guilty then he confesses his
crime (¬(G → C)). From this statement, one can deduce
that the suspect is guilty in classical reasoning. Indeed, the
pair h{¬(G → C)}, Gi is a classical argument, since we
have ¬(G → C) ≡ ¬(¬G ∨ C) ≡ G ∧ ¬C. This comes
from the interdefinability of connectives in classical logic
and the double-negation elimination. However, the previous
pair is not a constructive argument, since we are not able to
construct G from ¬(G → C).
∆0 , C, D ` A ∨ B
∆0 , C ∧ D ` A ∨ B
[∧L ]
where ∆ = ∆0 , A ∧ B, by applying the induction hypothesis
on ∆0 , C, D ` A ∨ B, we obtain a proof of ∆0 , C, D ` A or
∆0 , C, D ` B in GIPL . Hence, we have a proof of ∆0 , C ∧
D ` A or ∆0 , C ∧ D ` B in GIPL . Therefore, (∆, A) is a
constructive argument or (∆, B) is a constructive argument.
In the case of [→L ]:
Disjunction Property
The disjunction property is one of the most important properties satisfied in intuitionistic logic. It says that if a formula
A ∨ B is valid, then A is valid or B is valid. This property is
not satisfied in classical logic. Indeed, the formula p ∨ ¬p is
valid without p and ¬p being individually valid in classical
logic. From the point of view of constructivism, the disjunction property says that to construct the object A ∨ B, it is
necessary to be able to construct at least one of the objects
A and B.
Here we use the fact that intuitionistic logic enjoys the
disjunction property to show that if a constructive argument
has a support ∆ which does not contain the disjunction connective, and a conclusion of the form A ∨ B, then from ∆
we can construct one of the formulas A and B, i.e., h∆, Ai
or h∆, Bi is a constructive argument. In order to show this
property, we use a proof system for IPL in the sequent calculus formalism-style.
Let us recall that an inference rule has the following form:
∆0 , C → D ` C
∆0 , D ` A ∨ B
∆0 , C → D ` A ∨ B
[→L ]
where ∆ = ∆0 , A → B, by applying the induction hypothesis on ∆0 , D ` A ∨ B, we obtain a proof of ∆0 , D ` A or
∆0 , D ` B in GIPL . Hence, using the rule [→L ], we have a
proof of ∆0 , C → D ` A or ∆0 , C → D ` B in GIPL . Therefore, h∆, Ai or h∆, Bi is a constructive argument.
Note that Proposition 6 is not satisfied in classical logic.
For instance, the pair h{¬(p ∧ q)}, ¬p ∨ ¬qi is a classical
argument where the support does not contain the disjunction
connective. However, neither h{¬(p ∧ q)}, ¬pi nor h{¬(p ∧
q)}, ¬qi are classical arguments.
P1 · · · Pn
[R]
C
Computing Constructive Arguments using
Modal Logic S4
where [R] is its name, C its conclusion and P1 , . . . , Pn its
premises. An axiom can be seen as a rule without premises.
A proof system is defined as a set of inference rules. Proofsearch in a sequent calculus corresponds to a bottom-up construction of derivations using its inference rules, i.e., a construction from the conclusion to axioms.
We consider here the sequent calculus GIPL for IPL described in Figure 1 (see (Troelstra and Schwichtenberg
In this section we borrow from classical argumentation
framework the characterization of classical arguments in
terms of minimal inconsistent sets. As shown in (Besnard et
al. 2010) this characterization offers nice tractability properties for computing classical arguments. We provide such
a characterization for constructive arguments using modal
logic S4 (Blackburn, de Rijke, and Venema 2001).
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[Id]
p → r, q → r, p ` p
[Id]
q → r, p, r ` r
[Id]
p → r, q → r, q ` q
[Id]
p → r, q, r ` r
[→L ]
[→L ]
p → r, q → r, p ` r
p → r, q → r, q ` r
[∨L ]
p → r, q → r, p ∨ q ` r
[→R ]
p → r, q → r ` p ∨ q → r
Figure 2: A proof in GIPL
Modal Logic S4
in the same way as in (Besnard et al. 2010). However, notice that the computation of minimal inconsistent subsets in
modal logics is much less studied than in classical propositional logic. An interesting idea would be exploring how the
MUS computation methods in classical propositional logic
could benefit to modal logics. This is left for future work.
The set of S4 formulæ is obtained by extending the propositional language with the modal connectives et ♦:
A ::= p | ⊥ | A ∧ A | A ∨ A | A → A | A | ♦A.
Similarly to intuitionistic logic, modal logic S4 has a possible world semantics where we use a universe of worlds
with an accessibility relation between worlds which is reflexive and transitive. More precisely, an S4 model is a
triple M = (W, , V ), where is a preorder over W and
V : W → 2Prop is an interpretation. Hence, a S4 model
can be seen as a Kripke model of intuitionistic logic without
Kripke monotonicity property.
The forcing relation, denoted S4
M , is inductively defined on
formula structure as follows:
S4
w S4
M p iff p ∈ V (w); w M ⊥ never holds;
Conclusion and Future works
Constructivism is an approach which requires that to prove
the existence of an object, it is necessary to be able to
construct it. So far the main application of intuitionistic
logic in computer science is using the Curry-Howard correspondence (Howard 1980) which corresponds to a direct relationship between constructive proofs and computer
programs (Nordström, Petersson, and Smith 1990; PaulinMohring and Werner 1993).
In this paper we show the benefits of using intuitionistic logic to reason about inconsistency in argumentation theory. In particular Dung’s framework is instantiated with this
logic. In this setting, not only does the support of an argument deduce the conclusion of that argument but also constructs that conclusion. The present paper comes to complete existing works studying the validity of the logic-based
instantiations of Dung’s framework (Amgoud and Besnard
2013). While the focus of these works has been on the quality of the output of the logic-based argumentation frameworks (in terms of postulates), no attention has been paid on
the argument itself, in particular the support of the argument.
In our setting, a set of formulas which deduces a conclusion
would not be a support of an argument for that conclusion if
the reasons for such a deduction are not exactly identified.
For example h{¬(O ∧ T )}, ¬O ∨ ¬T i is not an argument in
our setting while h{O, ¬(O ∧ T )}, ¬O ∨ ¬T i is. In addition,
we provided a characterization of being a constructive argument in terms of a minimal inconsistent subset using Gödel’s
embedding of intuitionistic logic into the modal logic S4.
Our work should be useful in diagnosis-based applications
and law reasoning, to cite few.
As a future work, we intend to investigate the use of the
Curry-Howard correspondence in encoding the proofs of the
constructive arguments. Indeed, we know that each proof of
a constructive argument can be encoded as a λ-term in a
typed λ-calculus (Howard 1980). In this context, we plan to
consider a constructive argumentation framework where we
associate to each constructive argument a λ-term encoding
a method used in the argument to construct its conclusion
from its support. In this case, one of the perspectives consists
in defining a new type of attack relations over the λ-terms.
S4
S4
w S4
M A ∧ B iff w M A and w M B;
S4
S4
w S4
M A ∨ B iff w M A or w M B;
S4
S4
w S4
M A → B iff if w M A then w M B;
0
0
0 S4
w S4
M ♦A iff ∃w ∈ W s.t. w w and w M A;
0
0
0 S4
w S4
M A iff ∀w ∈ W with w w , w M A;
Embedding Intuitionistic Logic into S4
We describe here Gödel’s embedding of intuitionistic logic
into modal logic S4 (see, e.g., (Troelstra and Schwichtenberg 1996)). Intuitively, this embedding comes from the fact
that the Kripke models of intuitionistic logic are S4 models.
The definition of the embedding (·)S4 is by induction on formula structure as follows:
(p)S4 = p; (⊥)S4 = ⊥; (A ∧ B)S4 = (A)S4 ∧ (B)S4 ;
(A ∨ B)S4 = (A)S4 ∨ (B)S4 ;
(A → B)S4 = ((A)S4 → (B)S4 ).
We have the following property:
Proposition 7. Γ `i C is valid in intuitionistic logic iff
(Γ)S4 ` (C)S4 is valid in S4.
Hence, since the logical consequence concept in S4 is
classical, we obtain the following proposition:
Proposition 8. The pair h∆, Ci is a constructive argument
iff (∆)S4 ∪{¬(C)S4 } is a minimal inconsistent subset in S4.
Prop. 8 states that the use of S4 allows to provide a simple characterization of being a constructive argument similar to that of being a classical argument. Such a characterization can be used in a constructive argument generation
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